稀疏加组稀疏优化问题的方向稳定点及其光滑化方法

Directional Stationary Points of Sparse plus Group Sparse Optimization Problems with Its Smoothing Methods

摘要: 本文研究稀疏加组稀疏优化问题的非凸松弛模型，其中惩罚项既含有稀疏惩罚，又含有组稀疏惩罚，对稀疏惩罚和组稀疏惩罚均采用折叠凹惩罚函数进行连续松弛，得到复合非光滑非凸优化模型。为刻画此模型的最优性条件，给出了其方向导数刻画和方向稳定点，分析了方向稳定点的特征及其局部最优性质。为计算模型的方向稳定点，构造了模型的光滑化逼近问题，并证明了光滑化问题的一阶稳定点收敛于模型的方向稳定点In this paper, we study the nonconvex relaxation model for the sparse plus group sparse optimiza-tion problem, in which the penalty term contains both sparse penalty and group sparse penalty. As continuous relaxations, the folded concave penalty functions are used to relax both sparse penalty and group sparse penalty, which results the compound nonsmooth and nonconvex optimization model. In order to characterize the optimality of the nonconvex relaxation problem, the directional derivative and the directional stationary point are introduced, and then the characteristics of the directional stationary points and its local optimality are analyzed. To calculate the directional sta-tionary points of the relaxation model, the smoothing approximation problem is constructed, and it is proved that the stationary points of the smoothing problem converge to the directional stationary point of the relaxation problem, which provides a theoretical guarantee for the calculation of the directional stationary point of the relaxation problem by using the smooth methods.，为使用光滑方法计算模型的方向稳定点提供了理论保证。